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How can the dual martingale help solving the primal optimal stopping problem?

Aurélien Alfonsi, Ahmed Kebaier, Jérôme Lelong

TL;DR

The paper addresses variance in primal optimal stopping pricing by exploiting a purely dual martingale approximation $\hat M$ as a control variate within a Longstaff–Schwartz framework. It formalizes how to apply LS to $Z-\hat M$, and optionally use a dual-based construction to produce a policy, with a careful three-sample scheme to control bias. Numerical results across Put, Butterfly, Basket, Max-Call, and Min-Butterflies Bermudan options show variance reductions spanning from tens to orders of magnitude, and the effective coefficient $\lambda$ often lies near 1 when $\hat M$ closely approximates the true $M^*$. The findings demonstrate that even an approximate dual martingale can markedly improve pricing and hedging efficiency for multi-asset Bermudan options, offering a practical pathway to more accurate and stable estimators.

Abstract

Motivated by recent results on the dual formulation of optimal stopping problems, we investigate in this short paper how the knowledge of an approximating dual martingale can improve the efficiency of primal methods. In particular, we show on numerical examples that accurate approximations of a dual martingale efficiently reduce the variance for the primal optimal stopping problem.

How can the dual martingale help solving the primal optimal stopping problem?

TL;DR

The paper addresses variance in primal optimal stopping pricing by exploiting a purely dual martingale approximation as a control variate within a Longstaff–Schwartz framework. It formalizes how to apply LS to , and optionally use a dual-based construction to produce a policy, with a careful three-sample scheme to control bias. Numerical results across Put, Butterfly, Basket, Max-Call, and Min-Butterflies Bermudan options show variance reductions spanning from tens to orders of magnitude, and the effective coefficient often lies near 1 when closely approximates the true . The findings demonstrate that even an approximate dual martingale can markedly improve pricing and hedging efficiency for multi-asset Bermudan options, offering a practical pathway to more accurate and stable estimators.

Abstract

Motivated by recent results on the dual formulation of optimal stopping problems, we investigate in this short paper how the knowledge of an approximating dual martingale can improve the efficiency of primal methods. In particular, we show on numerical examples that accurate approximations of a dual martingale efficiently reduce the variance for the primal optimal stopping problem.
Paper Structure (12 sections, 2 theorems, 38 equations, 1 figure, 5 tables)

This paper contains 12 sections, 2 theorems, 38 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction and framework
  2. The dual martingale as a control variate for the primal problem
  3. Methodology
  4. Numerical examples
  5. One dimensional options
  6. Bermudan options on many assets
  7. Basket option
  8. Max-call option
  9. Min Butterflies
  10. A proxy for the optimal stopping policy
  11. Variance decomposition in the Longstaff Schwartz algorithm
  12. Calculation of $\hat{M}$: the algorithm

Key Result

Proposition 3.1

Let $\hat{M}$ be a ${\mathcal{F}}$-martingale and $\tau=\inf\{n \ge 0: \hat{U}_n=Z_n \}$. Then, we have $\tau= \hat{\tau}_0$, $Z_{\tau}-\hat{M}_{\tau}=\max_{0\le n\le N} Z_n-\hat{M}_n$ and

Figures (1)

  • Figure 1: Histogram of $\hat{\tau}_0-\hat{\tau}^{LS_1,Q_2}$ on the Put option example obtained with $50000$ samples, $P=50$ and $Q_2=50000$, using the martingale $\hat{M}$ obtained with $\bar{N}=20$ subticks (left, with $Q_1=2\times 10^6$) or using the martingale $\hat{M}$ that includes the European call option (right, with $Q_1=10^5$), see Equation \ref{['def_mg_incr']}.

Theorems & Definitions (3)

  • Proposition 3.1
  • Proof 1
  • Theorem B.1